Let $\{X_i\}_{0\leq i\leq\infty}$ be i.i.d. random vectors in $\mathbb{R^d}$. I would like to show that the probability of one point being in the convex hull of the others goes to one with the number of points: $$\tag{1}\label{claim} \lim_{n\to\infty}\mathbb{P}(X_0\in\mathrm{Conv}(X_1, \ldots, X_n)) = 1, $$ where the probability is taken with respect to the realization of $(X_0, \ldots, X_n)$.

While this seems intuitive, the following is a counterexample: in $\mathbb{R}^2$, if the $X_i$ are uniformly sampled from the circle $\mathbb{S}^1$, every sampled point almost surely creates a new vertex of the convex hull, and therefore $\mathbb{P}(X_0\in\mathrm{Conv}(X_1, \ldots, X_n)) = 0$.

If the $(X_i)$ are sampled from a continuous distribution, is my initial claim $\eqref{claim}$ true?

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